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Bimodal distribution
s with the same variance but different means. The figure shows the probability density function (p.d.f.), which is an average of the bell-shaped p.d.f.s of the two normal distributions.]] In statistics, a bimodal distribution is a continuous probability distribution with two different modes. These appear as distinct peaks (local maxima) in the probability density function, as shown in Figure 1. Terminology When the two modes are unequal the larger mode is known as the major mode and the other as the minor mode. The least frequent value between the modes is known as the antimode. The difference between the major and minor modes is known as the amplitude. In time series the major mode is called the acrophase and the antimode the batiphase. Examples Examples of variables with bimodal distributions include the time between eruptions of certain geysers, the color of galaxies, the size of worker weaver ants, the age of incidence of Hodgkin's lymphoma, the speed of inactivation of the drug isoniazid in US adults, the absolute magnitude of novae, and the circadian activity patterns of those crepuscular animals that are active both in morning and evening twilight. Important bimodal distributions include the arcsine distribution and the beta distribution. Mixture distributions A bimodal distribution most commonly arises as a mixture of two different unimodal distributions (i.e. distributions having only one mode). In other words, the bimodally distributed random variable X is defined as Y with probability \alpha or Z with probability (1-\alpha), where Y'' and ''Z are unimodal random variables and 0 < \alpha < 1 is a mixture coefficient. For example, the bimodal distribution of sizes of weaver ant workers shown in Figure 2 arises due to existence of two distinct classes of workers, namely major workers and minor workers. In this case, Y'' would be the size of a random major worker, ''Z the size of a random minor worker, and α'' the proportion of worker weaver ants that are major workers. A mixture of two normal distributions has five parameters to estimate: the two means, the two variances and the mixing parameter. A mixture of two normal distributions with equal standard deviations is bimodal only if their means differ by at least twice the common standard deviation. Estimates of the parameters is simplified if the variances can be assumed to be equal (the homoscedastic case). Mixtures of other distributions require additional parameters to be estimated. A mixture of two unimodal distributions with differing means is not necessarily bimodal. The combined distribution of heights of men and women is sometimes used as an example of a bimodal distribution, but in fact the difference in mean heights of men and women is too small relative to their standard deviations to produce bimodality. Bimodal distributions have the peculiar property that - unlike the unimodal distributions - the mean may be a more robust sample estimator than the median.Mosteller F, Tukey JW (1977) Data analysis and regression: a second course in statistics. Reading, Mass, Addison-Wesley Pub Co This is clearly the case when the distribution is U shaped like the arcsine distribution. It may not be true when the distribution has one or more long tails. Moments of mixtures Let : f( x ) = p g_1( x ) + ( 1 - p ) g_2( x ) where ''g''i is a probability distribution and ''p is the mixing parameter. The moments of f''(x) areKim T-H, White H (2003) On more robust estimation of skewness and kurtosis: Simulation and application to the S & P 500 index : \mu = p \mu_1 + ( 1 - p ) \mu_2 : \nu_2 = p[ \sigma_1^2 + \delta_1^2 ] + ( 1 - p )[ \sigma_2^2 + \delta_2^2 ] : \nu_3 = p [ S_1 \sigma_1^3 + 3 \delta_1 \sigma_1^2 + \delta_1^3 ] + ( 1 - p )[ S_2 \sigma_2^3 + 3 \delta_2 \sigma_2^2 + \delta_2^3 ] : \nu_4 = p[ K_1 \sigma_1^4 + 4 S_1 \delta_1 \sigma_1^3 + 6 \delta_1^2 \sigma_1^2 + \delta_1^4 ] + ( 1 - p )[ K_2 \sigma_2^4 + 4 S_2 \delta_2 \sigma_2^3 + 6 \delta_2^2 \sigma_2^2 + \delta_2^4 ] where : \mu = \int{ x f( x ) dx } : \delta_i = \mu_i - \mu : \nu_r = \int{ ( x - \mu )^r f( x ) dx } and ''S''i and ''K''i are the skewness and kurtosis of the ''i''th distribution. Multimodality More generally, a '''multimodal distribution' is a continuous probability distribution with two or more modes, as illustrated in Figure 3. Summary statistics Bimodal distributions are a commonly used example of how summary statistics such as the mean, median, and standard deviation can be deceptive when used on an arbitrary distribution. For example, in the distribution in Figure 1, the mean and median would be about zero, even though zero is not a typical value. The standard deviation is also larger than deviation of each normal distribution. Ashman's D A statistic that may be useful is Ashman's D:Ashman KM, Bird CM, Zepf SE(1994) Astronomical J 108: 2348 : D = 2^\frac{ 1 }{ 2 } \frac{ | \mu_1 - \mu_2 | }{ \sqrt{ ( \sigma_1^2 + \sigma_2^2 ) } } where μ''1, ''μ''2 are the means and ''σ''1 ''σ''2 are the standard deviations. For a mixture of two normal distributions ''D > 2 is required for a clean separation of the distributions. Bimodality index The bimodality index assumes that the distribution is a sum of two normal distributions with equal variances but differing means.Wang J, Wen S, Symmans WF, Pusztai L, Coombes KR (2009) The bimodality index: a criterion for discovering and ranking bimodal signatures from cancer gene expression profiling data. Cancer Inform 7:199-216 It is defined as follow: : \delta = \frac{ \mu_1 - \mu_2 }{ \sigma } where μ''1, ''μ''2 are the means and ''σ is the common standard deviation. : BI = \delta \sqrt{ p( 1 - p ) } where p'' is the mixing parameter. Bimodality coefficient Sarle's bimodality coefficient ''b isEllision AM (1987) Effect of seed dimorphism on the density-dependent dynamics of experimental populations of Atriplex triangularis (Chenopodiaceae). Am J Botany 74(8): 1280-1288 : \beta = \frac{ \gamma^2 + 1 }{ \kappa } where γ'' is the skewness and ''κ is the kurtosis. The kurtosis is here defined to be the standardised fourth moment around the mean. The value of b'' lies between 0 and 1.Pearson K (1916) Mathematical contributions to the theory of evolution, XIX: Second supplement to a memoir on skew variation. Phil Trans Roy Soc London. Series A 216 (538–548): 429–457. Bibcode 1916RSPTA.216..429P. doi:10.1098/rsta.1916.0009. JSTOR 91092 The formula for a finite sample isHellwig B, Hengstler JG, Schmidt M, Gehrmann MC, Schormann W, Rahnenführer J (2010) Comparison of scores for bimodality of gene expression distributions and genome-wide evaluation of the prognostic relevance of high scoring genes. BMC Bioinformatics 11:276 : b = \frac{ g^2 + 1 }{ k + 3 ( 1 - \frac{ ( n - 1 )^2 }{ ( n - 2 )( n - 3 ) } ) } where ''n is the number of items in the sample, g'' is the sample skewness and ''k is the sample kurtosis. The value of b'' for the uniform distribution is 5/9. This is also its value for the exponential distribution. Values greater than 5/9 may indicate a bimodal or multimodal distribution. The maximum value (1.0) is reached only by a Bernoulli distribution with only two distinct values or the sum of two different Dirac delta functions. The distribution of this statistic is unknown. It is related to a statistic proposed earlier by Pearson - the difference between the kurtosis and the square of the skewness (''vide infra). Statistical tests Unimodal vs bimodal distribution A necessary but not sufficient condition for a symmetrical distribution to be bimodal is that the kurtosis be less than three.Gneddin OY(2010) Quantifying Bimodality.Muratov AL, Gnedin OY (2010) Modeling the metallicity distribution of globular clusters. Ap J (submitted) arXiv:1002.1325 Here the kurtosis is defined to be the standardised fourth moment around the mean. The reference given prefers to use the excess kurtosis - the kurtosis less 3. Pearson in 1894 was the first to devise a procedure to test whether a distribution could be resolved into two normal distributions.Pearson K (1894) Contributions to the mathematical theory of evolution: On the dissection of asymmetrical frequency-curves. Phil Trans Roy Soc Series A, Part 1, 185: 71-90 This method required the solution of a ninth order polynomial. In a subsequent paper Pearson reported that for any distribution skewness2 + 1 < kurtosis.Pearson K (1916) Mathematical contributions to the theory of evolution, XIX: Second supplement to a memoir on skew variation. Phil Trans Roy Soc London. Series A 216 (538–548): 429–457. Bibcode 1916RSPTA.216..429P. doi:10.1098/rsta.1916.0009. JSTOR 91092 Later Pearson showed thatPearson K (1929) Editorial note. Biometrika 21: 370-375 : b_2 - b_1 \ge 1 where b''2 is the kurtosis and ''b''1 is the square of the skewness. Equality holds only for the two point Bernoulli distribution or the sum of two different Dirac delta functions. These are the most extreme cases of bimodality possible. The kurtosis in both these cases is 1. Since they are both symmetrical their skewness is 0 and the difference is 1. Baker proposed a transformation to convert a bimodal to a unimodal distribution.Baker GA (1930) Transformations of bimodal distributions. Ann Math Stat 1 (4) 334-344 Haldane suggested a test based on second central differences.Haldane JBS (1951) Simple tests for bimodality and bitangentiality. Ann Eugenics 16 (1) 359–364 DOI: 10.1111/j.1469-1809.1951.tb02488.x To test whether a univariant distribution is unimodal or bimodal, Larkin introduced a test based on the F test.Larkin RP (1979) An algorithm for assessing bimodality vs. unimodality in a univariate distribution. Behavior Research Methods 11 (4) 467-468 DOI: 10.3758/BF03205709 Later Benett instead used a G test.Bennett SC (1992) Sexual dimorphism of ''Pteranodon and other pterosaurs, with comments on cranial crests. J Vert Paleont 12 (4) 422-434 Tokeshi proposed another test for bimodality.Tokeshi M (1992) Dynamics and distribution in animal communities; theory and analysis. Researches in Population Ecology 34:249–273Barreto S, Borges PAV, Guo Q (2003) A typing error in Tokeshi’s test of bimodality. Global Ecology & Biogeography 12: 173–174 General tests To test if a distribution is other than unimodal, several additional tests have been devised: the bandwidth test,Silverman BW (1981) Using kernel density estimates to investigate multimodality. J Roy Statist Soc Ser B 43:97-99 the dip test,Hartigan JA, Hartigan PM (1985) The dip test of unimodality. Ann Statist 13 (1) 70-84 the excess mass test,Mueller DW, Sawitzki G (1991) Excess mass estimates and tests for multimodality. JASA 86, 738 -746 the MAP test,Rozál GPM Hartigan JA (1994) The MAP test for multimodality. J Classification 11 (1) 5-36 DOI: 10.1007/BF01201021 the mode-existence test,Minnotte MC (1997) Nonparametric testing of the existence of modes. Ann Statist 25 (4) 1646-1660 the runt test,Hartigan JA, Mohanty S (1992) The RUNT test for multimodality. J Classifcation 9: 63-70Andrushkiw RI, Klyushin DD, Petunin YI (2008) Theory Stoch Processes 14 (1) 1-6 the span test,Hartigan JA (1988) The span test of multimodality and the saddle test. See also * Overdispersion References Category:Probability distributions